EML 5060 Analysis in Mechanical Engineering 10/20/00
Closed book Van Dommelen 10:45-11:35am

Show all reasoning and intermediate results leading to your answer, or credit will be lost. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator.

1.

We want to drill a hole in a body with surface

x³y - yz + xz -xy = 0

at the point with coordinates (1,1,1). The drill should hit the surface straight on. Find the axis of the drill in terms of x, y, and z only. Solution.

2.

Consider a vibrating string. Assume we measure the transverse displacement of the string at various times at 6 equally spaced points (including the end points.) In particular we measure the six displacements to be:

• (0,1,1, 1,1,0) at time t=0
• (0,3,3, 1,1,0) at time t=1
• (0,4,2,-2,0,0) at time t=2
• (0,3,1,-1,1,0) at time t=3
• (0,6,4, 0,2,0) at time t=4

Of course, the string displacements are sums of independent natural modes with amplitudes that vary with time. Based on the measured displacements, what can you say about the number of modes that are significant in the vibration? Solution.

3.

The average magnetic field vector in a region is given by . Use Gram-Schmidt to find an orthogonal coordinate system in which is aligned with the positive x'-axis. Solution.